A SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES. Snježana Maksimović, Svetlana Mincheva-Kamińska, Stevan Pilipović, and Petar Sokoloski

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1 PUBLICATIONS DE L INSTITUT MATHÉMATIQUE Nouvelle série, tome 1(114) (16), DOI: 1.98/PIM161417M A SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES Snježana Maksimović, Svetlana Mincheva-Kamińska, Stevan Pilipović, and Petar Sokoloski Abstract. We introduce and investigate two types of the space U of s-ultradistributions meant as equivalence classes of suitably defined fundamental sequences of smooth functions; we prove the existence of an isomorphism between U and the respective space D of ultradistributions: of Beurling type if = (p! t ) and of Roumieu type if = {p! t }. We also study the spaces T and T of t-ultradistributions and t-ultradistributions, respectively, and show that these spaces are isomorphic with the space S of tempered ultradistributions both in the Beurling and the Roumieu case. 1. Introduction That distributions based by Sobolev [9] and Schwartz [8] on functional analysis can be founded on a more elementary sequential approach was remarked by Mikusiński already in [18] and [19]. This idea was accomplished by him in cooperation with Sikorski in [,1] and then, in the extended form, together with the third author Antosik in [1]. Roumieu and Beurling in [7] and [] introduced two types of ultradistribution spaces, substantially larger than the space of distributions. However only the famous papers [1 14] of Komatsu which substantially extended the knowledge on the structure of these spaces gave impulse to an intensive development of the theory of ultradistributions of both types in various directions. In particular, the theory became an important tool of microlocal analysis. Similarly as in the case of distributions one can expect that an ultradistribution can also be viewed as, in a sense, a limit of a sequence of functions or, more precisely, as an equivalent class of sequences of smooth functions, suitably approximating it. Our aim in this paper is to provide a sequential approach to the theory of non-quasi-analytic ultradistributions of both Beurling and Roumieu types [1]. Analogously to the sequential theory of distributions [1], we introduce sequential 1 Mathematics Subject Classification: Primary 46F5; Secondary 46F1. Key words and phrases: fundamental sequences, Hermite expansions. Communicated by Gradimir Milovanović. 17

2 18 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI ultadistributions, called shortly s-ultadistributions, as equivalence classes of fundamental sequences of smooth functions which, however, are defined by means of ultradifferential operators instead of differential operators. The difference is intrinsic and requires distinct techniques: instead of polynomials we have to use functions of sub-exponential growth and apply their specific properties. In showing that the sequential approach is equivalent to the classical approach to ultradistribution theory (see [1 14]) one needs to know intrinsic structures of ultradistributions as well as of tempered ultradistributions (see [5 1, 17 5]); the equivalence of the two approaches will be proved through Hermite expansions and certain structural properties of tempered ultradistributions. In order to simplify our exposition we will consider only the Gevrey sequence of functions of the form M p = p! t (p Z + ) for t > 1; they satisfy all conditions usually assumed for a general sequence (M p ) p. Therefore we use the simplified symbols D (t) (Ω), D {t} (Ω) for the spaces D (Mp) (Ω), D {Mp} (Ω) of test functions on an open set Ω R d and S (t) (R d ), S {t} (R d ) for the spaces S (Mp) (R d ), S {Mp} (R d ) of test functions on R d, respectively. This concerns also their duals, i.e., D (t) (Ω), D {t} (Ω) are the spaces of ultradistributions of Beurling and Roumieu type on the set Ω and S (R (t) )d, S {t} (R d ) are the spaces of tempered ultradistributions of Beurling and Roumieu type on R d, respectively. We traditionally use the upper index for a common notation of the considered spaces both in the Beurling and Roumieu cases, i.e., D (Ω), S (R d ), D (Ω), S (R d ) are common symbols for the pairs of spaces listed above. The mentioned spaces were investigated in [5, 1, 1, 16, 5] and in many other papers. It should be noted that another approach to the theory of ultradistributions was developed by D. Vogt, R. Meise and their collaborators. There exists an extensive literature in this direction with many applications; we refer here just to a few of them (and references therein): [3,4,3,3]. Our approach to ultradistributions is similar to that presented in [1] for distributions. We begin with the definition of a special kind of fundamental sequences of smooth functions and the corresponding equivalence classes called s- ultradistributions which are elements of the space that we denote by U (Ω). This is done in sections and 3 together with an analysis of operations on s-ultradistributions, the convergence structure in U (Ω) and actions of s-ultradistributions on test functions belonging to D (Ω). In section 4 we introduce the spaces T and T of t-and t-ultradistributions, respectively. Again we discuss their structure, the convergence in them and actions of considered tempered ultradistributions on elements of the respective spaces of test functions. It is well-known (see e.g. [8]) that there exists a topological isomorphism between the space S (R d ) and the Köthe echelon space s of sequences of sub-exponential growth. Using this fact we prove in section 5 that the spaces T and T are topologically isomorphic with the space S (R d ). Applying the results of section 5, we prove in section 6 the existence of a sequential topological isomorphism between the spaces U (Ω) and D (Ω) Preliminaries. The sets of all positive integers, nonnegative integers, real and complex numbers are denoted by N, N, R and C, respectively.

3 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 19 For x = (x 1,..., x d ) R d, y = (y 1,..., y d ) R d, α = (α 1,..., α d ) N d, β = (β 1,..., β d ) N d and λ R, we use the following notation x + y := (x 1 + y 1,..., x d + y d ) R d ; x + λ := (x 1 + λ,..., x d + λ) R d ; λx := (λx 1,..., λx d ) R d ; x y if x j y j for j = 1,..., d; d ( ) α d ( ) x α := x αj j ; α! := α αj 1!... α d!; := for α β; β j=1 j=1 x := (x x d) 1/ ; α := α α d ; D α = Dx α := D α Dα d, where Dαj j := ( i / x j ) αj (j = 1,..., d) and the following summation notation d := ; α N d α β := β 1 α 1= β j β d α d = The symbol X for X R d means the interior of X and the symbol K V for an open V R d means that K is a compact subset of V. By Ω we denote a fixed open subset of R d. By C(R d ) and for K Ω we denote the sets of all continuous functions on R d and K, respectively, and by C(Rd ) and the uniform convergences on R d and K of sequences of functions in C(R d ) and, respectively; the latter is the convergence in the Banach space with the supremum norm. We denote a sequence (α n ) n N of numbers (functions, distributions, ultradistributions) shorter by (α n ) or (α n ) n and the mapping Ω x F(x) by F or F(x). The norm in L (R d ) is denoted by and the convergence in L (R d ) by. The support of a function (distribution, ultradistribution) f by supp f. A function f is called compactly supported if there is a K R d such that supp f K. For the Fourier transform of ϕ S(R d ) we use the two symbols: F(ϕ) = ϕ := R ϕ(x)e i x, dx; clearly, F(D α ϕ) = ξ α ϕ (ξ R d ). For d the properties of the spaces of test functions D(Ω), S(R d ) and their duals D (Ω), S (R d ) we refer to [8]. We recall some notions from [1]. By the associated function, corresponding to the Gevrey sequence (p! t ) p for a fixed t > 1, we mean the following function: M(ρ) := sup p N log + ρ p /p! t = e kρ1/t for ρ >, where k > is an appropriate constant. Denote by R the set of all sequences (r p ) of positive numbers strictly increasing to infinity. By the (r p )-associated function, corresponding to (r p ) R, we mean the function: N (rp)(ρ) := sup p N log + ρ p /N p for ρ >, where (N p ) p N is defined by means of (r p ) as follows p (1.1) N := 1; N p := p! t R p, where R p := r j, for p N. Note that if (r p ) R, then for every k > there is a ρ > such that N (rp)(ρ) e kρ1/t for ρ > ρ (see [1]). j=1.

4 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI Let K Ω and h >. We recall the definitions of some spaces of test functions [1] { E t,h (K) := ϕ C D α ϕ(x) } (Ω): P h,k (ϕ) := sup < ; h α α! t D t,h K D (t) K D {t} K x K,α N d := Et,h (K) {ϕ C (Ω) : supp ϕ K}; := lim h := lim h D t,h K ; D t,h K ; D(t) (Ω) := lim D (t) K ; K Ω D{t} (Ω) := lim D {t} K. K Ω As already said, we use the common symbol D (Ω) for the spaces D (t) (Ω) and D {t} (Ω) and D (Ω) for their duals. Recall now (see [5]) the definitions of the spaces S (t) (R d ) and S {t} (R d ), invariant under the Fourier transform S t h(r d ) := { f S(R d ): C > α, β N d S (t) (R d ) := lim S h t (Rd ); h S {t} (R d ) := lim S h t (Rd ). h x α β f } h α+β α! t β! C ; t Notice that the space S (t) (R d ) is nontrivial if t > 1/, while S {t} (R d ) is nontrivial if t 1/. The spaces S (t) (R d ) and S {t} (R d ) are denoted commonly by S (R d ) and their duals by S (R d ). The Hermite polynomials H n and the corresponding Hermite functions h n are defined on R by H n (x) := ( 1) n e x( d dx ) n(e x ), h n (x) := ( n n! π) 1 e x / H n (x) for x R, n N. The d-dimensional Hermite functions h n are defined by h n (x) := h n1 (x 1 )... h nd (x d ), x = (x 1,..., x d ) R d, n N d. They form an orthonormal basis for L (R d ) and are the eigenfunctions of the product H = d i=1 ( / x i +x i ) of the one-dimensional Hermite harmonic oscillators, so that H α = d i=1 ( / x i + x i and )αi H α h k (x) = (k + 1) α h k (x) = d (k i + 1) αi h k (x), x R d, α, k N d. i=1 Note that H is a self-adjoint operator. For f S(R d ), the Hermite coefficients are c k = R d fh k = (f, h k ) L for k N d. We have D (R d ) S (R d ) S(R d ) L (R d ) S (R d ) S (R d ) D (R d ), where the symbol means that the identity mapping is a continuous and dense embedding.

5 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 1 A sequence (δ n ) of the form δ n := n d ϕ(n ), n N, where ϕ D (R d ), ϕ = 1 in B(, 1/) and ϕ = out of B(, 1) (B(x, r) denotes the closed ball with the center at x and radius r) is called a delta sequence in D (R d ). 1.. Ultradifferential operators. We recall the definitions and some results related to ultradifferential operators from [1 16]. A formal expression P (D) = a αd α (a α C), corresponding to the function P (z) = a αz α (z C d ), is called an ultradifferential operator of the Beurling class (p! t ) resp. of the Roumieu class {p! t }) if it satisfies the condition h > C > (resp. h > C > ) α N d a α Ch α (α!) t ; in the Roumieu case, the condition can be expressed in the equivalent form (r p ) R C > α N d a α C (α!) t R α, where R α is defined in (1.1) and t > 1 will be fixed throughout the paper. We use the common term ultradifferential operator of the class in both cases of Beurling and Roumieu. If P (D) is an ultradifferential operator of the Beurling class (resp. of the Roumieu class), then the function P (z) satisfies, by [1, Proposition 4.5], the estimate h > C > (resp. h >, C > ) z C d P (z) Ce h z 1/t ; in the Roumieu case, the estimate can be written in the equivalent form (r p ) R C > ξ R d P (ξ) Ce c( ξ )1/t, where c is the subordinate function of (r p ), i.e., an increasing function on [, ) such that c() = and c(ρ)/ρ as ρ corresponding to (r p ) by means of the identity: M(c(ρ)) = N (rp)(ρ) (ρ > ), where N (rp) is the (r p )-associated function (see [1]). We denote by P (t) (resp. P {t} ) the class of ultradifferential operators P r (D) of Beurling type (resp. P (rp)(d) of Roumieu type) of the form (1.) (1.3) P r (D) = P (rp)(d) = ( 1 + ( 1 + d j=1 d j=1 D j D j ) l p=1 ) l p=1 [ ( d 1 + [ 1 + j=1 ( d j=1 D j D j )/r p t ] ( = )/r p pt ] ( = a p D ), p p= b p D ), p where r >, (r p ) R and l. Replacing D j by ξ j in (1.) and (1.3) we get the ultra-polynomials P r (ξ) and P (rp)(ξ) of the Beurling and Roumieu type corresponding to the ultradifferential operators P r (D) and P (rp)(d), respectively. They can be described in the following way (see [1]): p=

6 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI Ultra-polynomials of Beurling type are of sub-exponential growth, i.e., there are constants C 1, C, C > and h 1, h, h > such that (1.4) C 1 e h1 ξ 1/t P r (ξ) C e h ξ 1/t, ξ R d, and a p Ch p /p! t, p N. The description of ultra-polynomials in the Roumieu case is more difficult. One can prove, similarly to the Beurling case, that for a given (r p ) R and its subordinate function c there exists a constant C > such that (1.5) Ce c( ξ )1/t P (rp)(ξ), ξ R d. To get a suitable upper estimate we have to find a sequence (r,p ) R and its subordinate function c such that the inequality (1.6) (1 + ξ ) l P (rp)(ξ) C e c( ξ )1/t, ξ R d. holds for some C > and all l. For this aim we use the following property of a subordinate function which is a consequence of Lemma 3.1 (see also Lemma 3.1) in [1]. Let c be an arbitrary subordinate function and put c := c. There exists a sequence (r p ) R such that the (r p )-associated function N (r p ) and the subordinate function c corresponding to (r p) satisfy the inequality Consequently, M( c(ρ)) N (r p )(ρ) = M(c (ρ)), ρ >, c (ρ) c(ρ) = c(ρ), ρ >. The above remarks can be formulated as follows: Lemma 1.1. For an arbitrary subordinate function c, corresponding to some sequence (r p ) R, and h > there exist a sequence (rp ) R and its subordinate function c such that c (ρ) hc(ρ), ρ >. In particular, for a given subordinate function c there exists another subordinate function c (both corresponding to appropriate sequences from R) such that c (ρ) c(ρ) for all ρ >. In Subsection 4. we will need also the following lemma. Lemma 1.. (a) If P r P (t) (resp. P (rp) P {t} ), then there exist r > (resp. (r p) R), C > and ε > such that D α P r (x) Cα! 1/t er x, x R d, α N d ε α ( resp. D α P (rp)(x) Cα! ) ε α ec (r p ) ( x )1/t, x R d, α N d, where c (r p ) is the subordinate function corresponding to the sequence (r p ).

7 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 3 (b) If P r P (t) (resp. P ( rp) P {t} ), then there exist r > (resp. ( r p) R), C > and ε > such that D α (1/P r (x)) Cα! 1/t e r x, x R d, α N d ε α ( resp. D α (1/P ( rp)(x)) Cα! 1/t, x R d, α N) d, ε α e c ( r p) ( x ) where c ( r p ) is the subordinate function corresponding to the sequence ( r p). Proof. In the proof of both parts we use Lemma 1.1 and the Cauchy integral formula for ultra-polynomials on the circles K(x, ε) around x R d. The proof of (b) follows from the estimate D α (1/P ( rp)(x)) Cα! ε α e N ( rp)( x ), x R d, α N d, shown in [6, Lemma.1], because we can construct c ( r p ) such that N ( rp)( x ) = M(c ( rp)( x )) = Cc ( rp)( x ) 1/t = c ( r p )( x ) 1/t for some C > and x >, in view of Lemma 3.1 in [1] and Lemma 1.1. The proof of (a) follows from Proposition 4.5 in [1]; see also the last part of the proof of Theorem 1. in [1]. The symbol P will be common for the classes P (t) and P {t} of ultradifferential operators of Beurling and Roumieu types and P will mean the space of t-ultradistributions of both types. The corresponding spaces of ultradifferentiabile functions will be denoted by Pu and P u, respectively. This notation looks complicated but it helps to distinct the different use of P : P (D), P (x), P (ξ). To simplify the exposition we will usually consider ultradifferential operators of the form (1.) and (1.3), but in some proofs we need their general form. Denote by µ β the operator acting on measurable functions G as follows: (µ β G)(ξ) := (iξ) β G(ξ) for ξ R d and β N d ; in particular µ G = G. We will use in the sequel the following assertion: for arbitrary β N d, q [1, ] and P r (D) P (t) with r > (resp. P (rp)(d) P {t} with (r p ) R) there exists P r (D) P (t) with r > r (resp. P ( rp) P {t} with ( r p ) R, r p / r p as p ) such that (1.7) ( resp. µ β P (rp) P ( rp) µ β P r P r L q (R d ) and F 1( µ β P r P r ) L q (R d ) and F 1( µ β P (rp) P ( rp) L q (R d ) ) ) L q (R d ) and, analogously, with Pu instead of Pu, the following one ( Pr (α + 1) )α Zd ( ( P(rp)(α (1.8) P r (α + 1) + 1) ) lq resp. P ( rp)(α + 1) lq), α Z d where P (α + 1) := k = a k d i=1 (α i + 1) ki for α N d. The following well-known assertions will also be used in the sequel; their proofs can be found e.g. in [5,5].

8 4 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI Lemma 1.3. (a) A smooth function ϕ on R d belongs to S (R d ) iff for arbitrary P P and P 1 Pu we have P 1P ( D)ϕ <. (b) If ϕ n S (R d S ) (n N ) and ϕ n ϕ as n, then for arbitrary P P and P 1 Pu S we have P 1 P ( D)ϕ n P 1 P ( D)ϕ as n. (c) If ϕ n S (R d S ) (n N ) and ϕ n ϕ as n, then for every P P S we have P (H)ϕ n P (H)ϕ as n.. Fundamental sequences Let us recall that Schwartz distributions in the sequential approach presented in [1] are equivalence classes of fundamental sequences of smooth functions defined with the use of derivatives of finite order. We introduce s-ultradistributions of Beurling and Roumieu type in a similar way, but our fundamental sequences are defined by means of the ultradifferential operators P r (D) and P (rp)(d), respectively, instead of finite order differential operators. If P P and F is an integrable function compactly supported, then P (z)f(f)(z) (z C d ) is an entire function of sub-exponential growth on R d. If the inverse Fourier transform F 1 (P F) is a locally integrable function, then we define (.1) P (D)F(x) := F 1 (P F)(x), x R d to give the meaning for the formal acting of the ultradifferential operator P (D) on a compactly supported smooth function. If F is a compactly supported smooth function such that supp F K 1 Ω and P P is of the form P (D) := a αd α, then the left hand side of (.1) is meant as follows (.) P (D)F(x) := lim k P k(d)f(x), x K, where P k (D) := k a αd α and the limit in (.) is assumed to exist for every x K and to be a smooth function on K. In this case the limit defines f(x) = P (D)F(x) for x K and gives the meaning of (.3) below. Definition.1. A sequence (f n ) of smooth functions defined on an open set Ω R d is called s-fundamental (of type, i.e., of Beurling or Roumieu type, respectively) in Ω if for arbitrary K 1 Ω and K K 1 there exist an ultradifferential operator P (D) P, a sequence (F n ) of smooth functions on Ω and a continuous function F on Ω such that (.3) f n = P (D)F n on K (n N), supp F n K 1 (n N ) and F n F as n. The equality in (.3) is meant in the sense of (.). In the sequel, for a given K Ω we will always take a set K 1 Ω with K K 1 which is sufficiently close to K, not referring explicitly about it (one can show, taking an appropriate cut-of function, that the definition does not depend on the choice of K 1 ).

9 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 5 Remark.1. 1 One can consider in (.3) all ultradifferential operators of class, not only belonging to P, which gives a more general form of the definition. Since both formulations are equivalent, we will use the above one for simplicity. Let Ω 1, Ω be open sets in R d such that Ω 1 Ω. If a sequence (f n ) is s-fundamental in Ω, then it is s-fundamental in Ω 1. 3 Let (f n ) be a sequence of smooth functions in Ω. If for every open set Ω Ω the sequence (f n Ω ) is s-fundamental in Ω, then (f n ) is s-fundamental in Ω. Definition.. Let (f n ) and (g n ) be s-fundamental sequences in an open set Ω. We write (f n ) (g n ) if for arbitrary K 1 Ω and K K 1, there exist an ultradifferential operator P P and sequences (F n ), (G n ) of smooth functions on Ω such that f n = P (D)F n, g n = P (D)G n on K (n N), supp F n, supp G n K 1 (n N), and F n G n as n, where the symbol F n G n means that (F n ) and (G n ) converge in to a common continuous function H on Ω. Remark.. It is clear that if (f n ) and (g n ) are s-fundamental sequences in Ω, then (f n ) (g n ) iff the sequence f 1, g 1, f, g, f 3, g 3,... is s-fundamental in Ω. Obviously, the relation is reflexive and symmetric. To prove its transitivity we need some auxiliary statements. Proposition.1. Fix K 1 Ω and K K1. Assume that (f n ) satisfies Definition.1 in the Roumieu case, i.e., f n = P (rp)(d)f n on K, supp F n K 1 for n N and F n F as n for a sequence (F n ) of smooth functions and a continuous function F on Ω. Then there are a P ( rp) P {t}, where ( r p ) R with r p / r p as p and F 1 (P (rp)/p ( rp)) L 1 (R d ), smooth functions F n (n N) and a continuous function F on Ω such that f n = P ( rp)(d) F n on K (n N), supp F n K 1 (n N ) and F n F as n. The same assertion holds in the Beurling case (with the corresponding notation). Proof. By (1.7), for arbitrary β N d (in particular, for β = ) we can find ( r p ) R with r p / r p as p such that P (rp) µ β L 1 (R d ) and F 1( P ) (rp) µ β L 1 (R d ). P ( rp) P ( rp) Define [ (.4) Fn := κ K F 1( P ) ] (rp) (κ K F n ), n N, P ( rp)

10 6 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI where κ K is a smooth function in D {t} (Ω) such that { 1, if x K, (.5) κ K (x) =, if x K1 c. Then supp F n K 1 for n N and [ P ( rp)(d) F n (x) = P ( rp)(d) F 1( P (rp) P ( rp) ) ] (κ K F n ) (x) = [ F 1 (P (rp)) (κ K F n ) ] (x) = P (rp)(d)(κ K F n )(x) = f n (x) for x K and n N. Since F 1( ) P (rp)/p ( rp) L 1 (R d ) and κ K F n κ K F as n, it follows that F n F as n. Proposition.. If for arbitrary K 1 Ω and K K1 there are a P P, smooth functions F n (n N) and a continuous function F on Ω with supp F n K 1 (n N ) such that F n F and P (D)F n (x) for x K as n, then F = on Ω. In particular, if F is a smooth function on Ω and P (D)F(x) = for x Ω, then F = on Ω. Proof. Fix K 1 Ω, K K 1 and κ K as in (.5). By the assumption, (.6) P (D)(κ K F n )(x) (x K), κ K F n κ K F as n. Since lim m [P (D) P m (D)](κ K F n ) = in R d, it follows from (.6) that and lim n lim P m(ξ) κ K F n (ξ) = lim P (ξ) κ K F n (ξ) =, ξ R d. m n lim P (ξ)[ κ K F n (ξ) κ K F (ξ) ] =, ξ R d. n This implies P (ξ) κ K F (ξ) = for ξ R d, so κ K (x)f (x) = for x K. Hence F = in K and thus F = in Ω, since K Ω was arbitrarily chosen. The particular case is clear if we take F n = F for n N. Proposition.3. Fix K, K 1 Ω such that K K1. Assume that, for sequences (r p ), ( r p ) R, smooth functions F n, Fn on Ω (n N) and continuous functions F, F on Ω, we have and f n = P (rp)(d)f n on K (n N), supp F n K 1 (n N ), F n F as n f n (x) = P ( rp)(d) F n on K (n N), supp F n K 1 (n N ), F n F as n.

11 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 7 Then there are a sequence ( r p ) R, r p / r p, r p / r p as p such that (.7) F 1( P ) (rp), F 1( P ) ( rp) L 1 (R d ), P ( rp) P ( rp) and sequences (F n,1 ), (F n, ) of smooth functions and continuous functions F,1, F, on Ω, with supp F n,1, supp F n, K 1 (n N ), such that (.8) f n = P ( rp)(d)f n,j on K (n N), F n,j F,j as n for j = 1,. Moreover F,1 = F, in Ω. The same assertion also holds in the Beurling case (with the corresponding notation). Proof. The existence of ( r p ) R satisfying (.7) follows from (1.7). Define [ F n,1 := κ K F 1( P ) [ (rp) (κ K F n ) ], F n, := κ K F 1( P ) ] ( rp) (κ K Fn ), P ( rp) P ( rp) where κ K is a smooth function in D {t} (Ω) which satisfies (.5). Using Proposition.1, one can deduce (.8). Finally, by Proposition., we conclude that F,1 = F, in Ω. Proposition.4. Relation introduced in Definition. is transitive. Proof. We will prove the assertion only in the Roumieu case; the proof in the Beurling case is similar. Suppose that (f n ) (g n ) and (g n ) (h n ) and fix K, K 1 Ω so that K K 1. Now select K Ω such that K K and K K 1. By the assumption and Definition., there exist (r p ), ( r p ) R and sequences (F n ), (G n ), ( G n ), (H n ) of smooth functions on Ω such that and f n = P (rp)(d)f n, g n = P (rp)(d)g n on K (n N), supp F n, supp G n K (n N), F n G n as n g n = P ( rp)(d) G n, h n = P ( rp)(d)h n on K (n N), supp G n, supp H n K 1 (n N), Gn C( K) H n as n. In view of Proposition.3, there exist an appropriate ( r p ) R and convergent sequences (F n,1 ), (G n,1 ), ( G n,1 ), (H n,1 ) of smooth functions, all having supports contained in K 1, such that f n = P ( rp)(d)f n,1, g n = P ( rp)(d)g n,1 = P ( rp)(d) G n,1, h n = P ( rp)(d)h n,1 on K. Moreover F n,1 G n,1 and G n,1 H n,1 as n. If we put now H n,1 (x) := G n,1 (x) G n,1 (x) + H n,1 (x), x K, then h n = P ( rp)(d) H n,1 in K and F n,1 H n,1 as n, which means that (f n ) (h n ). This completes the proof.

12 8 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI.1. Sequential ultradistributions. Definition.3. Let (f n ) be a s-fundamental sequence (of type ) in an open set Ω R d. The class of all s-fundamental sequences equivalent to (f n ) with respect to the relation is called a sequential ultradistributon or, shortly, s-ultradistribution (of type ) and denoted by f = [f n ]. The set of all s-ultradistributions (of type ) on Ω is denoted by U (Ω). Remark.3. 1 By Proposition., f = [f n ] = on Ω for f U (Ω) if for arbitrary K 1 Ω and K K 1 there exist a sequence (F n) of smooth functions on Ω and an ultradifferential operator P P such that f n = P (D)F n on K (n N), supp F n K 1 (n N) and F n as n. Let f U (Ω). If f = on Ω 1 for every Ω 1 Ω, then f = on Ω. Definition.4. By the support of an s-ultradistribution f U (Ω) we mean the complement of the union of all open sets where f =. We say that an s- ultradistribution f = [f n ] or a s-fundamental sequence (f n ) is compactly supported if there exists K R d such that supp f n K for n N. Then we write supp f K or supp (f n ) K. In this case there exist a sequence of smooth functions (F n ), a continuous function F, an ultradifferential operator P P and K 1 Ω so that f n (x) = P (D)F n (x) on R d (n N), supp F n K 1 (n N ) and F n F as n. Example.1. Let F be a compactly supported continuous function in R d, (δ n ) be a delta sequence in D (R d ) and let F n := F δ n for n N. Then (F n ) is a s-fundamental sequence on R d. Example.. Let (f n ) be a s-fundamental sequence on Ω R d and (K n ) be an increasing sequence of compact sets such that K n Kn+1 for n N. Put Ω := n N K n and consider the open sets Ω n := {x Ω: d(x, Ω) > 1/n}, and functions κ n C (Ω) such that κ n (x) = Ω n,n := {x Ω n : d(x, Ω n ) > 1/n} { 1, if x Ω n,n,, if x Ω c n, for n N. Then the sequence ( f n ), where { ((κ n f n ) δ n )(x), if x Ω n, f n (x) =, if x Ω c n, is s-fundamental on Ω and (f n ) ( f n ).

13 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 9 Example.3. Let f D (Ω) for an open Ω R d, let Ω n, Ω n,n and κ n be as in Example. and let P (rp) P {t}. Put { F 1 ( f n /P (.9) F n (x) := (rp))(x), if x Ω,, if x Ω c n, for n N, where f n := (κ n f) δ n = P (rp)(d)f n and P (rp) means in (.9) the function corresponding to an ultra-differential operator P (rp). Since κ n f, for every n N, is bounded by a polynomial, it follows from (1.6) that (f n ) is a s-fundamental sequence of the Roumieu type on Ω, that is [f n ] U {t} (Ω). In a similar way, we can also represent f as an element of U (t) (Ω) by the use P r (D) instead of P (rp)(d). Remark.4. If (f n ) is a s-fundamental sequence in the sense of Definition.1, then F n F as n for every K Ω. Examples.1 and.3 show that every s-fundamental sequence (f n ) for which (.3) holds can be identified with the formal representation f = P (D)F on K, since from the general theory of ultradistributions we know that for arbitrary K 1 Ω and K K1 there exist P P and F C(Ω) such that f = P (D)F on K and supp F K 1. This will be justified by (3.3) and the last section... Operations on s-ultradistributions. We start from the operations of addition and multiplication by a constant. Let f, g U (Ω) and λ C, where f = [f n ], g = [g n ] for some s-fundamental sequences (f n ), (g n ). Using Proposition.3, one can prove that (f n + g n ) and (λf n ) are s-fundamental sequences on Ω, so we may define s-ultradistributions f + g := [f n + g n ] and λf := [λf n ]. By Remark., the definitions are consistent. Consequently, U (Ω) is a vector space. Next consider the operation of differentiation. If f = [f n ] U {t} (Ω), i.e., (f n ) is a s-fundamental sequence of the Roumieu type, and let β N d. We will show that the sequence (f n (β) ) is s-fundamental of the Roumieu type. For arbitrary K 1 Ω and K K1 take (F n) and P (rp)(d) according to Definition.1. Since f n (β) = P (rp)(d)f n (β) on K, it follows from (1.7) as in the proof of Proposition.1 for β = that there exist P ( rp) P {t}, a sequence ( F n ) of smooth functions and a continuous function F on Ω defined by (cf. (.4)) F n := κ K [F 1( P ) ] (rp) µ β (κ K F n ), n N, P ( rp) with supp F n K 1 (n N ), such that P (rp)(d)f n (β) = P ( rp)(d) F n on K and F n F as n. Consequently, (f n (β) ) is s-fundamental and we define f (β) = [f n (β) ]. By Remark., the definition is consistent and f (β) U {t} (Ω). An analogous assertion holds in the Beurling case. Let us discuss now the operations of multiplication and convolution by a function from E (Ω). We consider only the Roumieu case. The Beurling case is similar.

14 3 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI Fix ω E {t} (Ω) and let f = [f n ] U {t} (Ω). We will show that the sequence (ωf n ) is s-fundamental, so one can define ωf := [ωf n ] E {t} (Ω) and the definition is consistent, by Remark.. For every K Ω there exist P (rp) P {t}, a sequence (F n ) of smooth functions and a continuous function F in Ω with supp F n K 1 (n N ) such that ωf n = ωp (rp)(d)f n in K and F n F, n. We can assume that ω is compactly supported multiplying it by a cut-of function equal to 1 on K. We have ω(ξ) Ce h ξ 1/t and P (rp)(ξ) F n C 1 e c( ξ )1/t, ξ R d for some constants C >, C 1 >, h > and a subordinate function c, in view of (1.6). By (1.7), there exists a P ( rp) P {t}, where ( r p ) R with r p / r p as p, such that ( ω (P (rp) F n ))/P ( rp) L 1 (R d ). Now, defining ( ω F ) G n := κ K F 1 (P(rp) n ), n N, P ( rp) we have ωf n = P ( rp)(d)g n on K (n N), supp G n K 1 (n N ) and G n G as n. Hence, (ωf n ) is a s-fundamental sequence on Ω. Moreover, if f = [f n ] and ω D {t} (Ω), then ω f = [ω f n ], since ω P (D)F n = P (D)(ω F n ) on every compact set K R d for n N. More generally, we have the following assertion: if (f n ) and (g n ) are s-fundamental sequences on R d and supp(g n ) K R d, then (f n g n ) is a s- fundamental sequence on R d. 3. Sequences of s-ultradistributions Definition 3.1. Let f m U (Ω) for m N, i.e., f m = [(fn m) n], where (fn m ) n means a s-fundamental sequence representing f m for m N. We say that the sequence (f m ) converges to f in U (Ω) and write f m s f as m or s-lim m f m = f if for arbitrary K 1 Ω and K K1 there exist a P P, smooth functions Fn m on Ω (m N, n N) and continuous functions F m on Ω (m N ), all supported by K 1, such that f m n = P (D)F m n on K (n N, m N ); F m n F m n F n as m uniformly in n N; F m as n (m N ) and F m F as m. We know that the above assumptions imply that lim lim F m m n n = lim lim F m n m n in. Theorem 3.1. If the limit s-lim m f m exists, then it is unique. Proof. Assume that f m s f and f m s g, where f = [f n ], g = [g n ] U (Ω) and f m = [(f m n ) n ] U (Ω) for m N. We will prove that f = g.

15 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 31 Fix arbitrary K 1 Ω and K K 1. According to Definition 3.1, there exist P, P P, smooth functions F m n, F n, G m n, G n, on Ω and continuous functions F m, F, G m, G on Ω (n, m N), all supported by K 1, such that f m n F m n F m n = P (D)F m n, f n = P (D)F n on K (n, m N); F n as m uniformly in n N; F n F as n ; F m as n (m N); F m F as m ; and, on the other hand, f m n G m n G m n = P(D)G m n, g n = P (D)G n on K (n, m N); G n as m uniformly in n N; G n G as n ; G m as n (m N); G m G as m. By Proposition.3, there exist an ultradifferential operator P P, smooth functions F m n, Ḡm n, Fn, Ḡ n and H m n := F m n Ḡm n, H n := F n Ḡn on Ω as well as continuous functions F m, Ḡm, F, Ḡ and H m := F m Ḡm, H := F Ḡ on Ω (n, m N), all supported by K 1, such that = P (D)H m n, f n g n = P(D)H n on K (n, m N); H m n H m n H n as m uniformly in n N; H n G as n ; H m as n (m N) and H m H as m. Hence Hn m = on K for n, m N, by Proposition.. This implies that H =, i.e., F = Ḡ on K. Since K Ω was fixed arbitrarily, we conclude that f = g on Ω Action on test functions from D (Ω). Let f = [f n ] U (Ω), where (f n ) is a s-fundamental sequence satisfying Definition.1, i.e., for arbitrary K, K 1 Ω with K K 1 there exist P (D) P, smooth functions F n (n N) and a continuous function F on Ω such that (3.1) f n = P (D)F n on K (n N), supp F n K 1 (n N ) and F n F as n. By the action of f on the test functions from D (Ω) we mean the mapping (3.) D (Ω) ϕ (f, ϕ) U (Ω) R, where (3.3) (f, ϕ) U (Ω) := lim n If, beside (3.1), we have K f n (x)ϕ(x)dx = K F (x)[p ( D)ϕ](x)dx. f n = P (D) F n on K (n N), supp F n K 1 (n N ) and F n F as n

16 3 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI for some P (D) P, smooth functions F n (n N) and a continuous function F on Ω, then lim f n (x)ϕ(x)dx = F (x)[ P ( D)ϕ](x)dx, n K K i.e., definition in (3.3) is consistent. Clearly, 3. is a linear mapping. To prove that mapping (3.) is sequentially continuous we need the following result from [1]: if ϕ n ϕ in D (Ω), then P (D)ϕ n P (D)ϕ in D (Ω) for every P (D) P. Theorem 3.. (a) Let f U (Ω) and ϕ m ϕ as m in D (Ω). Then (f, ϕ m ) U (Ω) (f, ϕ ) U (Ω) as m. (b) Let f m f as m in U (Ω). Then (f m, ϕ) U (Ω) (f, ϕ) U (Ω) as m for every ϕ D (Ω). Proof. (a) Supose that f = [f n ], where (f n ) is a s-fundamental sequence, i.e., (3.1) holds for given K 1 Ω and K K1 and suitable P (D) P and F n (n N ). By (3.3), we have lim [(f, (ϕ m ϕ ) U m (Ω)] = lim F (x)[p ( D)(ϕ n ϕ )](x)dx =. m (b) Let ϕ DK. Using the notation of Definition 3.1 we obtain Since Fn m have lim (f m f, ϕ) U m (Ω) = lim lim m K n (f m n = lim m lim n K f n, ϕ) U (Ω) (F m n F n)(x)[p ( D)ϕ](x)dx. F n as m uniformly in n N and F n F as n, we lim lim (Fn m F n )(x)[p ( D)ϕ](x)dx =. n m K 4. Tempered sequential ultradistributions 4.1. t-tempered sequential ultradistributions. Recall that d H α = ( / x i + x i ) αi, α = (α 1,..., α d ) N d i=1 and that the symbol P means either P (t) or P {t}. Let P P. We will use ultradifferential operators of the form P (H) = a αh α of Beurling class (p! t ) (resp. of Roumieu class {p! t }) such that h > C > (resp. h > C > ) α N d in the Roumieu case the given condition is equivalent to (r p ) R C > α N d C a α (α!) t, R α where R α is defined in (1.1). a α Ch α (α!) t ;

17 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 33 Definition 4.1. A sequence (f n ) of functions from L (R d ) C (R d ) is called t-fundamental (of type ) in R d if there exist an ultradifferential operator P P, functions F n L (R d ) C (R d ) of the form F n = c α,nh α with (c α,n ) α N d l for n N and a function F L (R d ) such that (4.1) f n = P (H)F n on R d and F n F as n, where P (H)F n in (4.1) is meant in L (R d ) C (R d ) as follows k lim P k(h)f n = lim a α H α F n = c α,n P (α + 1)h α, k k (n N). We will need later the following assertion which enables us to change in representations of t-fundamental sequences an L -convergent sequence with a sequence which converges both in L (R d ) and uniformly on R d. Lemma 4.1. Assume that F n L (R d ) C (R d ) for n N and F n F as n. Denote F n := F 1 (G n ), where G n (ξ) := (1+ ξ ) d/ Fn (ξ) for ξ R d and n N. Then ( F n ) is a bounded sequence of smooth functions such that F n F as n and F C(R d ) n F as n. Proof. It is clear that the sequence ( F n ) is bounded and F n F, due to the Parseval identity. By the Schwarz inequality, we have F n F ( ) 1/ = F 1 (G n G ) (1 + ξ ) d dξ Fn F, R d which proves the uniform convergence. Definition 4.. Let (f n ) and (g n ) be t-fundamental sequences. We write (f n ) 1 (g n ) if there exist sequences (F n ), (G n ) of functions F n, G n L (R d ) C (R d ) (n N), both convergent in L (R d ), and an operator P P such that f n = P (H)F n, g n = P (H)G n on R d and F n G n as n. The following two assertions will be used in the sequel. Proposition 4.1. If there exist P P, functions F n L (R d ) C (R d ) for n N and F L (R d ) such that F n = c α,nh α with (c α,n ) α N d l for n N and, moreover, F n F and P (H)F n as n, then F = on R d. In particular, if F L (R d ) C (R d ) and P (H)F = in L (R d ), then F = on R d. Proof. By the assumption, we have (c α,n ) α N d (c α, ) α N d in l and P (H)F n = P (α + 1)c α,n h α as n. Consequently, (c α,n ) α N d in l as n. Hence F =. The particular case is clear.

18 34 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI Proposition 4.. Assume that f n = P (rp)(h)f n = P ( rp)(h) F n on R d (n N) and F n F, Fn F as n for some P (rp), P ( rp) P {t} and functions F n, F n L (R d ) C (R d ) and F, F L (R d ) of the form F n = c α,n h α, Fn = c α,n h α (n N ), where (c α,n ) α N d, ( c α,n ) α N d l for n N. Then there are a P ( rp) P {t}, where ( r p ) R with r p / r p and r p / r p as p, such that ( ) ( ) P(rp)(α + 1) P( rp)(α (4.) l + 1), l, P ( rp)(α + 1) P ( rp)(α + 1) α N d α N d and functions G n, G n L (R d ) C (R d ) and G, G L (R d ) of the form P (rp)(α + 1) G n = P ( rp)(α + 1) c P ( rp)(α + α,nh α, Gn = P ( rp)(α + 1) c α,nh α for n N, satisfying the conditions and f n = P ( rp)(h)g n = P ( rp)(h) G n on R d (n N) (4.3) G n G, Gn G as n. Moreover, G n = G n on R d for n N. The same holds in the Beurling case with an appropriate notation. Proof. The existence of P ( rp)(h) follows from (1.8). It is clear that H β h α = (α + 1) β h α for α, β N d. By 4., we have P (rp)(α + 1) G n = P ( rp)(α + 1) c c α,n h α α,nh α = P (rp)(h) P ( rp)(α + 1) for n N and a similar representation holds for G n (n N ). Hence c α,n h α P ( rp)(h)g n = P ( rp)(h)p (rp)(h) P ( rp)(α + 1) = P (rp)(h)p ( rp)(h) c α,n h α P ( rp)(α + 1) = P (r p)(h)f n = f n and, similarly, P ( rp)(h) G n = f n on R d forn N. We deduce from (4.) that G n and G n are smooth L functions and (4.3) holds. By Proposition 4.1, we conclude that G n = G n on R d for n N and, consequently, G = G. It is clear that the relation 1 is reflexive and symmetric. We shall prove that 1 is transitive. Proposition 4.3. Relation 1 is transitive.

19 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 35 Proof. We prove the assertion in the Roumieu case; the proof in the Beurling case is similar. Let (f n ) 1 (g n ) and (g n ) 1 (h n ). Then there exist P (rp), P ( rp) P {t} and sequences (F n ), (G n ), (G 1 n), (H n ) of functions in L (R d ) C (R d ), all convergent in L (R d ), such that f n = P (rp)(h)f n, g n = P (rp)(h)g n = P ( rp)(h)g 1 n, h n = P ( rp)(h)h n on R d for n N and F n G n, G 1 n H n as n. By Proposition 4., there is a P ( rp) P {t}, where ( r p ) R with r p / r p and r p / r p as p, and there are suitable functions F n, Ḡn, Ḡ1 n, H n on R d such that f n = P ( rp)(h) F n, g n = P ( rp)(h)ḡn = P ( rp)(h)ḡ1 n, h n = P ( rp)(h) H n on R d for n N and F n Ḡn, Ḡ1 n H n as n. Putting Φn := Ḡn Ḡ1 n+ H n, we have h n = P ( rp)(h)φ n on R d and, moreover, Fn Φ n as n. Hence (f n ) 1 (h n ), i.e., 1 is transitive. Definition 4.3. Let (f n ) be a t-fundamental sequence (of type ) in an open set Ω R d. The class of all t-fundamental sequences equivalent to (f n ) with respect to the relation 1 is called a t-tempered sequential ultradistributon or, shortly, t-ultradistribution (of type ) and denoted by f = [f n ]. The set of all t-ultradistributions (of type ) in Ω R d is denoted by T = T (R d ). Remark 4.1. As in the space U (Ω) (see Section.) we can consider appropriate operations in T. But we do not go into details, remarking only that the operations of addition and multiplication by a constant are well defined in this set, i.e., T is a vector space. Example 4.1. Let F L (R d ) and let (δ n ) be a delta-sequence in D (R d ). Define F n := F δ n for n N. Then (F n ) is a sequence of smooth functions in L (R d ) which is t-fundamental in both the Beurling and Roumieu cases. Example 4.. Let f = [f n ] T be of the form f n = P (H)F n, where P P and F n L (R d ) C (R d ) for n N. Moreover, assume that F n F as n for some function F L (R d ). Define f n := P (H)(F n δ n ) for n N. Then ( f n ) is a t-fundamental sequence in R d and (f n ) 1 ( f n ). Definition 4.4. Let f m T for m N, i.e., f m = [(f m n ) n ], where (f m n ) n means a t-fundamental sequence representing f m for m N. We say that the sequence (f m ) converges to f in T and write f m t f as m or t- lim m f m = f if there exist a P P, smooth functions F m n L (R d ) (m N, n N) and functions F m L (R d ) (m N ) such that f m n = P (H)F m n on R d (n N, m N ); F m n F m n Fn as m, uniformly in n N; F m as n (m N ) and F m F as m. Assumptions in the definition imply that lim lim F m m n n = lim lim F m n m n in L (R d ).

20 36 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI Theorem 4.1. If the limit t-lim m f m exists, then it is unique. Proof. In the Roumieu case, let f m s f and f m s g, where f m = [(f m n ) n ] T {t} for m N and f = [f n ], g = [g n ] T {t}. We will show that f = g. By Definition 4.4, there exist ultradifferential operators P (rp), P ( rp) P {t} with (r p ), ( r p ) R, smooth functions F m n, F n, G m n, G n L (R d ) and functions F m, F, G m, G L (R d ) (n, m N) such that f m n = P (rp)(h)f m n, f n = P (rp)(h)f n on R d (n, m N); F m n F m n and, on the other hand, f m n G m n G m n F n as m uniformly in n N; F n F as n ; F m as n (m N); F m F as m = P ( rp)(h)g m n, g n = P ( rp)(h)g n on R d (n, m N); G n as m uniformly in n N; G n G as n ; G m as n (m N); G m G as m. By Proposition 4., there exist a P ( rp) P {t}, where ( r p ) R, with r p / r p and r p / r p as p, smooth functions F m n, Ḡ m n, Fn, Ḡ n in L (R d ) and functions F m, Ḡm, F, Ḡ in L (R d ) for n, m N such that the following conditions are satisfied = P ( rp)(h)φ m n, f n g n = P (D)(H)Φ n on R d (n, m N); Φ m n Φ m n Φ n as m uniformly in n N; Φ n Φ as n ; Φ m as n (m N) and Φ m Φ as m, where Φ m n := F n m Ḡm n, Φ n := F n Ḡn, Φ m := F m Ḡm for n, m N and Φ := F Ḡ. Hence P ( r p)(h)φ n = lim m P ( rp)(h)(φ m n Φ n ) = on R d (n N). As in Proposition 4.1, we conclude that F = Ḡ on Rd and thus f = g. The proof in the Beurling case is similar. We need the following assertion: Lemma 4.. Let ϕ be a function in D (Ω) equal to 1 on B(, 1/) and let ϕ m (x) := ϕ(x/m) for x R d and m N. If f = [f n ] is a t-ultradistribution, then fϕ m t f as m. Proof. Let P P (F n ) be a sequence of functions corresponding to (f n ) according to Definition 4.1. Put Fn m := ϕ m F n for m, n N. Then f m = P (H)ϕ m F n T, because ϕ m F n ϕm F as n for every fixed m. Since F n ϕ (x/m) 1 F n (x) dx, n, m N, we have F m n F m n x > m F n as m uniformly in n. This completes the proof.

21 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 37 Corollary 4.1. For every t-ultradistribution f there exists a sequence (f m ) of t-ultradistributions such that f m t f as m. Remark 4.. As in the case of s-fundamental sequences commented in Remark.4, every t-fundamental sequence (f n ) for which (4.1) holds can be identified with the formal representation f = P (H)F, since any other representation defines the same element of T (see (4.15) of Subsection 4.3). 4.. t-tempered sequential ultradistributions. In this subsection we develop a sequential theory of tempered ultradistributions closely related to the sequential approach of sections and 3. Definition 4.5. A sequence (f n ) of smooth functions is called t-fundamental (of type ) in R d if there exist an ultradifferential operator P P, a function P 1 P u and functions F L (R d ) and F n L (R d ) C (R d ) for n N such that (4.4) f n = P (D)(P 1 F n ) on R d and F n F as n. The action of P (D) on P 1 F n is understood as in Section ; it is the limit of m a αd α (P 1 F n )(x) as m for x R d. The following assertion will enable us to transfer one form of a fundamental sequence into another one. For a given h > and a subordinate function c denote for simplicity E ±h (u) := e ±hu1/t and E ±c (u) := e ±c(u)1/t for u. Lemma 4.3. Let P 1, F n and F be as in (4.4). For P 1 assume (1.4) in the Beurling case (resp. (1.6) in the Roumieu case) in the form P 1 (x) CE h1 ( x ) (resp. P 1 (x) CE c1 ( x )), x R d, where h 1 > is a constant (resp. c 1 is a subordinate function). Then (a) for a given h > (resp. for a given subordinate function c) there exists r > (resp. (r p ) R) such that [Eh F 1 (P 1 /P r )](x) < (resp. [E c F 1 (P 1 /P (rp))](x) < ) for every x R d. (b) there exists r > (resp. (r p ) R) such that E h1 [ (P1 F n P 1 F ) F 1 (P 1 /P r ) ] ( resp. E c [ (P 1 F n P 1 F ) F 1 (P 1 /P (rp)) ] ) as n, where c is a subordinate function such that c 1/t 1 c 1/t. Proof. We will prove the assertions only in the Roumieu case; the proof in the Beurling case is similar. To prove (a) choose (r p) R such that E c ( x ) P (r p )(x) for x R d. The proof will be completed if we show that there exists (r p ) R such that

22 38 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI P (r p )(D)(P 1 /P (rp)) L 1 (R d ), since then the function P (r p )F 1 (P 1 /P (rp)) belongs to L (R d ). For all x R d, we have ( ) α (4.5) P (r p )(D)(P 1 /P (rp))(x) = a α [D α γ P 1 D γ (1/P (rp))](x), γ where C (4.6) a α (α!) t R α γ α ( R α := i α r i ), α N d for some C >. By Lemma 1., there exist a subordinate function c 1 (related to P 1 ) and a subordinate function c (rp) (suitably chosen to fulfill the inequality c 1 ( x ) c (rp)( x ) for x R d ) such that (4.7) D α γ C(α γ)! P 1 (x) ( x ), ε α γ x R d, α, γ N d, γ α; (4.8) D γ (1/P (rp))(x) Cγ! (rp), ε γ x R d, γ N d, By (4.5), (4.6), (4.7) and (4.8), we get P(r p )(D)(P 1 /P (rp))(x) C ( (/ε) α (α!) t 1 R α ) E c1 ( x )E c (rp) ( x ). This proves (a), since the sum on the right-hand side is finite. To prove (b) note that, by Lemma 4.1, we can assume that (F n ) is a bounded sequence of smooth functions in L (R d ). By the assumption and (1.5), there exists a suitable subordinate function c (depending on (r p ) R) satisfying (4.9) C := E 4c1 ( s )E c ( s )ds < R d and there is a constant C > such that [ (P 1 F n P 1 F ) F 1( P 1 /P (rp))] (x) C F n F ( R d E c1 ( x s )E c ( s )ds CC F n F E c1 ( x ) ) 1/ for all x R d, due to (4.9). Hence assertion (b) easily follows. Definition 4.6. Let (f n ) and (g n ) be t-fundamental sequences. We write (f n ) (g n ), if there exist sequences (F n ), (G n ) of functions F n, G n L (R d ) C (R d ) (n N), both convergent in L (R d ), an operator P P and a function P 1 Pu such that f n = P (D)(P 1 F n ), g n = P (D)(P 1 G n ) on R d and F n G n as n.

23 SEQUENTIAL APPROACH TO ULTRADISTRIBUTION SPACES 39 Proposition 4.4. If the assumptions of Definition 4.6 are satisfied for (f n ) and if P (D)(P 1 F n ), then F n as n. In particular, if P (D)(P1 F) =, then F =. Proof. Using the Fourier transform, we have P P 1 F n. The same is true for P 1 F n, then for P 1 F n and, finally, for F n. The particular case is clear. The key assertion in this subsection is related to the change of representative of some t-ultradistribution (see Definition 4.7 below). We consider the Roumieu case. Assume that P (rp), P ( rp) P {t}, P(r 1, P p) ( r p) P{t} u and (F n ), ( F n ) are sequences of functions in L (R d ) C (R d ), both convergent in L (R d ), satisfying Definition 4.5, i.e., (4.1) f n = P (rp)(d)(p 1 (r p) F n) = P ( rp)(d)(p ( r p) F n ) on R d (n N), so that (4.11) max { P (rp)(x), P ( rp)(x), P 1 (r p) (x), P ( r p) (x) } Ce c( x )1/t for a suitable subordinate function c and x R d. We may assume, without loosing generality, that P 1 (r p) = P ( r p) = P ( r p). Actually one can use in (4.1) instead of P 1 (r p) (x)f n(x) and P ( r p) (x) F n (x), the following expressions P ( rp)(x) P 1 (r p) (x)f n(x) P ( rp)(x) and P P ( r (x) F p) n (x) ( rp)(x), x R d, P ( rp)(x) respectively, where the sequence ( r p ) R is increasing slowly enough to guarantee L -convergence of the sequences ( P 1 (x)f ) ( (rp) n(x) P, (x) F ) ( rp) n (x). P ( rp)(x) P ( rp)(x) The above remarks concern the following proposition. Proposition 4.5. Assume that P (rp), P ( rp) P {t}, P ( rp) P u {t} and functions F n, F n L (R d ) C (R d ) for n N, and F, F L (R d ) satisfy Definition 4.5, i.e., f n = P (rp)(d)(p ( rp)f n ) = P ( rp)(d)(p F ( rp) n ) on R d (n N) and F n F, F n F as n, so that (4.11) holds. Then there exist P (r p ) P {t} with (rp ) R and functions F n L (R d ) C (R d ) for n N and F L (R d ) such that f n = P (r p )(D)(P F ( rp) n ) on R d and Fn F as n. Proof. We know that there exists (r p) R such that if we put G n := F 1 (F n /P (r p )) and Gn := F 1 ( F n /P (r p )) on R d (n N ), then f n = P (r p )(D)(P ( rp)g n ) = P (r p )(D)(P ( rp) G n ) on R d (n N) and, moreover, G n G and G n G as n.

24 4 MAKSIMOVIĆ, MINCHEVA-KAMIŃSKA, PILIPOVIĆ, AND SOKOLOSKI By Proposition 4.4, G n = G n on R d for n N, so the assertion follows for the functions F n := G n = G n (n N ). To prove that, introduced in Definition 4.6, is an equivalence relation, it suffices to show that it is transitive. Proposition 4.6. The relation is transitive. We omit the proof of the proposition, because it is similar to the proofs of Propositions.4 and 4.3. One has to use appropriate representations as it was demonstrated in those proofs. Definition 4.7. Let (f n ) be a t-fundamental sequence (of type ) in R d. The class of all t-fundamental sequences equivalent to (f n ) with respect to the relation is called a t-tempered sequential ultradistributon or, shortly, t-ultradistribution (of type ) and denoted by f = [f n ]. The set of all t-ultradistributions (of type ) in R d is denoted by T = T (R d ). We give the convergence structure in T. Definition 4.8. Let f m T for m N, i.e., f m = [(fn m) n], where (fn m) n means a t-fundamental sequence representing f m for m N. We say that the sequence (f m ) converges to f in T and write f m t f as m or tlim m f m = f if there exist P P and P 1 Pu, smooth functions F n m L (R d ) (m N, n N) and functions F m L (R d ) (m N ) such that f m n = P (D)(P 1 F m n ) on R d (n N, m N ); F m n F m n F n as m, uniformly in n N; F m as n (m N ) and F m F as m. The assumptions of the definition imply that lim lim F m m n n = lim lim F m n m n in L (R d ). By suitable modifications of the proofs of Proposition 4.4 and Theorem 4.1, one can prove the following theorem Theorem 4.. If the limit t-lim n f n exists, then it is unique. By [5] we know that every f S (R d ) can be identified with the formal representation f = P (D)(P 1 F ), where P P, P 1 Pu and F L (R d ) is a function of the form F = c α,h α L (R d ) with (c α, ) α N d l. Actually, we need the following assertion: Lemma 4.4. An f is an element of the space S (R d ) if and only if (4.1) f = P (D)(P 1 F ) for some P P, P 1 Pu and F L (R d ) of the form (4.13) F = c α, h α with (c α, ) l,